Summary: The strong chromatic number of a graph
Noga Alon
Abstract
It is shown that there is an absolute constant c with the following property: For any two
graphs G1 = (V, E1) and G2 = (V, E2) on the same set of vertices, where G1 has maximum
degree at most d and G2 is a vertex disjoint union of cliques of size cd each, the chromatic
number of the graph G = (V, E1 E2) is precisely cd. The proof is based on probabilistic
arguments.
1 Introduction
Let G = (V, E) be a graph on n vertices. If k divides n we say that G is strongly k-colorable if for
any partition of V into pairwise disjoint sets Vi, each of cardinality k precisely, there is a proper
k-vertex coloring of G in which each color class intersects each Vi by exactly one vertex. Notice
that G is strongly k-colorable if and only if the chromatic number of any graph obtained from G
by adding to it a union of vertex disjoint k-cliques (on the set V ) is k. If k does not divide n , we
say that G is strongly k-colorable if the graph obtained from G by adding to it k n/k - n isolated
vertices is strongly k-colorable. The strong chromatic number of a graph G , denoted by s(G), is
the minimum k such that G is strongly k-colorable. As observed in [6] if G is strongly k-colorable
then it is strongly k + 1- colorable as well, and hence s(G) is in fact the smallest k such that G
is strongly s-colorable for all s k.